Foundations of Teletraffic Science & Stochastic Queuing Models
1. The Foundations of Teletraffic Science: Agner Krarup Erlang & Poisson Processes
In the early twentieth century, Danish mathematician and engineer Agner Krarup Erlang (1878–1929), working for the Copenhagen Telephone Company (KTAS), published foundational papers that formalized teletraffic engineering and modern applied queuing theory. Erlang addressed a profound practical challenge confronting early telecommunications networks: determining the exact number of copper circuits or switchboard operators required to handle incoming telephone calls with an acceptable probability of call completion, while preventing prohibitive capital over-expenditure on idle plant infrastructure.
Traffic Intensity and the Erlang Unit (E): Standardized by the CCITT (now ITU-T) in 1946, the Erlang is a dimensionless measure of telecommunications traffic intensity. One Erlang represents the continuous, uninterrupted occupation of a single transmission circuit over an observation interval of one hour (3,600 seconds of cumulative call duration). Mathematically, traffic intensity A is derived from the mean call arrival rate λ and the average call holding duration h:
For example, an enterprise branch office experiencing 600 calls during its peak busy hour (λ = 600 calls/hr), with an Average Holding Time (AHT) of 180 seconds (3 minutes), generates:
A = (600 × 180) / 3600 = 108,000 / 3,600 = 30.0 Erlangs.
This means that at any random instant during the busy hour, an average of exactly 30 voice circuits are actively carrying conversation audio.
The Poisson Arrival Distribution: Erlang recognized that when telephone calls originate independently from a very large subscriber population, call arrivals follow a memoryless Poisson process. The probability P(k) that exactly k calls arrive within an arbitrary observation interval t is governed by:
In parallel, call holding durations h are modeled as exponentially distributed random variables with service rate parameter μ = 1/h. The fundamental hallmark of the exponential distribution is its memoryless property: the probability that an active call continues for an additional Δt seconds is strictly independent of how long the call has already been in progress. In Kendall notation, this memoryless arrival and service behavior is denoted by the letter M (Markovian).
2. Lost-Calls-Cleared (LCC) vs. Lost-Calls-Delayed (LCD) Models
The fundamental bifurcation in classical teletraffic engineering divides operational systems into two distinct queuing philosophies:
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Lost-Calls-Cleared (LCC) — Erlang B (M/M/m/m): When a call attempt arrives at a switching system and encounters all m transmission circuits busy, the call is instantly rejected and dropped. No waiting queue exists. The blocked call immediately departs the system with zero delay. Classical public switched telephone networks (PSTN), T1/E1 primary rate interfaces (PRI), and inter-office SIP trunks operate under this principle.
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Lost-Calls-Delayed (LCD) — Erlang C (M/M/m/∞): When an arriving call attempt encounters all m service agents or processing lines occupied, the caller enters a first-in, first-out (FIFO) queue of theoretically infinite buffer depth. The caller remains on hold until a server becomes free. Modern customer support desks, technical help desks, and inbound IVR contact centers operate under this delay model.
Kendall's Standard Queuing Notation: Queuing systems are classified using the standardized six-parameter descriptor A / B / c / K / N / D:
- A (Arrival Process): M for Poisson (exponential inter-arrival times), D for Deterministic, G for General distribution.
- B (Service Process): M for Exponential holding times, D for constant fixed service times, G for General arbitrary service times.
- c (Number of Servers): The count of parallel trunk circuits, radio channels, or contact center agents (e.g., 24 for a T1 trunk, 30 for an E1 trunk).
- K (System Capacity): Total allowable entities in system (servers + queue slots). For Erlang B, K = c (zero queue buffer). For Erlang C, K = ∞.
- N (Population Size): Number of potential call sources. Assumed infinite (∞) for Erlang B/C, or finite (S) for the Engset model.
- D (Queue Discipline): Defaulting to First-Come, First-Served (FIFO / FCFS).
The Retrial Phenomena — Extended Erlang B: In real-world telephony, human callers do not simply vanish upon hearing a fast-busy tone. A significant proportion of blocked subscribers (often 40% to 70%) immediately redial within seconds. The classical Erlang B model assumes zero retrials, which causes it to dangerously underestimate true blocking rates during network overload. Extended Erlang B (EEB) models subscriber recall probability R through an iterative convergence loop, mathematically inflating the raw offered load A into an expanded offered load A' until steady-state equilibrium is reached:
3. The Engset Model: When Infinite Source Assumptions Fail
Both Erlang B and Erlang C rely on the assumption of an infinite subscriber population (N → ∞). This assumption holds true for municipal telecommunications switches serving hundreds of thousands of homes. However, when dimensioning private branch exchanges (PBX) in enterprise offices, shipboard satellite communication links, or industrial radio concentrators where the number of active telephone handsets S is comparable to the number of available external lines m (specifically when the source-to-server ratio S / m < 10), Erlang B yields excessively pessimistic blocking estimates.
In a finite source environment, every subscriber who originates a call and occupies an active trunk is temporarily removed from the pool of idle callers. As trunk utilization increases, the arrival rate of new calls automatically drops. The arrival process is strictly state-dependent: when k lines are busy, the instantaneous call arrival rate is λk = (S - k) × γ, where γ is the calling rate per idle subscriber.
The Engset Formula accounts for this negative feedback mechanism using binomial state distributions:
Because idle callers cannot generate new calls while already engaged on a trunk line, the Engset model demonstrates that a finite user pool achieves a substantially lower blocking probability than predicted by Erlang B for identical nominal traffic levels.
4. VoIP & Modern Packet Voice Teletraffic Engineering
Modern telecommunications has largely transitioned from legacy time-division multiplexing (TDM) circuits (such as 64 kbps DS0 timeslots on T1/E1 PRIs) to packet-switched Voice over IP (VoIP) traversing SIP trunks and Carrier Ethernet backbones. While Erlang B remains the gold standard for determining the required number of simultaneous SIP sessions, network engineers must bridge teletraffic intensity to physical IP bandwidth (bits per second).
A single uncompressed VoIP call using the standard G.711 codec generates 64 kbps of raw pulse-code modulated (PCM) voice payload. However, transporting this audio over an IP packet network requires encapsulating the payload inside Real-time Transport Protocol (RTP), User Datagram Protocol (UDP), and Internet Protocol (IP) headers:
- RTP Header: 12 Bytes (sequence numbering, timestamps, synchronization source).
- UDP Header: 8 Bytes (source and destination ports, length, checksum).
- IPv4 Header: 20 Bytes (addressing, TTL, QoS/DSCP markings) or 40 Bytes for IPv6.
- Ethernet Framing: 14 Bytes MAC header + 4 Bytes 802.1Q VLAN tag + 4 Bytes CRC/FCS = 22 Bytes.
At the industry-standard packetization sample interval of 20 milliseconds (50 packets per second), each packet encapsulates:
Payload = 64,000 bps × 0.020 s = 1,280 bits = 160 Bytes.
Total Layer 3 packet size = 160 (Payload) + 12 (RTP) + 8 (UDP) + 20 (IP) = 200 Bytes.
Layer 3 Bandwidth = 200 Bytes × 8 bits/Byte × 50 packets/sec = 80.0 kbps per call.
With Layer 2 Ethernet framing (22 Bytes), the physical line rate is:
Physical Wire Rate = (200 + 22) × 8 × 50 = 88.8 kbps per concurrent call.
By contrast, a low-bitrate compressed codec like G.729 compresses speech to 8.0 kbps (20 Bytes payload per 20ms frame). When encapsulated with the same 40-byte IP/UDP/RTP overhead, the resulting Layer 3 packet size is 60 Bytes, consuming 24.0 kbps (or 32.8 kbps at Layer 2). Enabling Voice Activity Detection (VAD) and Comfort Noise Generation (CNG) leverages the statistical reality that human conversation involves one party listening while the other speaks, yielding approximately 35% statistical bandwidth savings across large multi-session SIP trunks.